Download Choosing Mesh Parameters for Complex Systems

Simplified Method of Modeling
Complex Systems of Simple Optics
using Wave-Optics:
An Engineering Approach
Dr. Justin D. Mansell, Steve Coy, Liyang Xu,
Anthony Seward, and Robert Praus
MZA Associates Corporation
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Outline
• Introduction
– Ray Matrices
– Lens Law
– Fourier Optics
•
•
•
•
Aperture Imaging Into Input Space
Finding the Field Stop & Aperture Stop
Mesh Determination for Complex Systems
Conclusions
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Ray Matrix Formalism
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Introduction - ABCD Matrices
• The most common
ray matrix
formalism is the
2x2 or ABCD that
describes how a
ray height, x, and
angle, θx, changes
through a system.
θx
x
A B
C D 


 x'  A B  x 
 '  C D   
 x
 x  
x '  Ax  B x
 x '  Cx  D x
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x’
θx’
2x2 Ray Matrix Examples
Propagation
θx
x
x '  x  x L
x’
 x '  1 L   x 
 '  0 1   
 x
 x  
Lens
x '  x  x / f
θx
 x'   1
 '   1/ f
 x  
θx’
x
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0  x 
1  x 
Example ABCD Matrices
Matrix Type
Propagation
Lens
Curved Mirror
(normal
incidence)
Curved
Dielectric
Interface
(normal
incidence)
Form
Variables
L = physical length
n = refractive index
f = effective focal length
1 L n 
0 1 


 1
 1 f

0
1
0
 1
 2 R 1


R = effective radius of
curvature
1
0

 n  n  R 1
2
1


n1 = starting refractive index
n2 = ending refractive index
R = effective radius of
curvature
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3x3 and 4x4 Formalisms
4x4
E = Offset
F = Added Tilt
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A B E  x 
C D F   

 x
 0 0 1   1 
3x3
• Siegman’s Lasers
book describes two
other formalisms: 3x3
and 4x4
• The 3x3 formalism
added the capability
for tilt addition and offaxis elements.
• The 4x4 formalism
included two-axis
operations like axis
inversion and image
rotation.
 Ax
C
 x



Bx
Dx
Ay
Cy
 x 
  
 x
By   y 
 
Dy   y 
5x5 Formalism
 Ax
C
 x
0

0
 0
• We use a 5x5 ray
matrix formalism as
a combination of
the 2x2, 3x3, and
4x4.
– Previously
introduced by
Paxton and Latham
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Bx
0
0
Dx
0
0
Ay
0
By
0
Cy
Dy
0
0
0
Ex   x 
Fx   x 
 
Ey   y 
 
Fy   y 
1   1 
Lens Law
• This equation
governs the
location of the
formation of an
image.
1 1 1
 
f d1 d 2
1
1 1 d1  f
f
  
 d2 
d1
d 2 f d1
fd1
d1  f
f  d1
M eff 
f
d 2  d1 M eff
object
image
f
d2
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d1
Huygens Principle
In 1678 Christian Huygens
“expressed an intuitive
conviction that if each point on
the wavefront of a light
disturbance were considered to
be a new source of a secondary
spherical disturbance, then the
wavefront at any later instant
could be found by constructing
the envelope of the secondary
wavelets.”
-J. Goodman, Introduction to
Fourier Optics (McGraw Hill,
1968), p. 31.
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Huygens-Fresnel Integral
1
U ( x2 , y2 )   U ( x1 , y1 )  h( x2 , y2 , x1 , y1 )  dx1  dy1
1 exp  jkr21 
h
cosn, r21 
j
r21
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2
Fresnel Approximation
• Fresnel found that
in modeling longer
propagations,
Huygens-Fresnel Integral
U ( x2 , y2 )   U ( x1 , y1 )  h( x2 , y2 , x1 , y1 )  dx1  dy1
h
– the cosine term
could be neglected
and
– the spherical term
could be
approximated by a
r2 term.
Fresnel Approximation
 r2 2 
exp( jkz)
U ( x2 , y 2 ) 
exp  jk  
jkz
 2z 

 r12 
U ( x1 , y1 )  exp  jk  

 2 z 

k


exp   j  x2 x1  y2 y1   dx1  dy1
z


where ra  xa  ya
2
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1 exp  jkr21 
cosn, r21 
j
r21
2
Notation Simplification
 r12 
Q1  exp  jk 
 2z 
Quadratic Phase Factor (QPF): Equivalent to the
effect a lens has on the wavefront of a field.
Fourier Transform
k


F (U )   U  exp   j x2 x1  y2 y1   dx1  dy1
z


exp( jkz)
P
Multiplicative Phase Factor: Takes into account the
jkz
overall phase shift due to propagation
U '  P  Q2  F U  Q1 
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Fresnel Approximation Validity
 a
z  16
 
from Siegman
4
1/ 3



Ch.16

1/ 3
 a 

z  
4  
from Goodman Ch. 4
4
spherical
parabolic
radius
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Fresnel Approximation Validity Examples
a (cm)
z >>
(π/4 a4/λ)1/3
Goodman
Nf max
100
92 m
10838
10
4.3 m
2335
1
0.2 m
503
0.1
9.2 mm
108
0.01
430 μm
23
0.001
20 μm
5
λ = 1.0 μm
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Implementing Fourier Propagation
• One-Step Fourier
Transform
U '  P  Q2  F U  Q1 
– Single Fourier integral
• Convolution
Propagator
U '   U  h  dx  dy
– Short distances
• Two Cascaded OneStep Propagations
U i  P  Qi  F U  Q1 
U '  P  Q2  F U  Qi  or
– Longer distances
U '  P  Q2  F Qi  F U  Q1  Qi 
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Discrete Sample Implementation
• When implementing Fourier
propagation on a computer, the
field is sampled at discrete
points.
• The mesh spacing between
samples (δ) and the number of
mesh points (N) required for
accurate modeling are
discussed later.
• The mesh spacing can be
different at the beginning of a
propagation (δ1) than at then
end (δ2)
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δ
Samples, Not Local Area Averages
• We represent the continuous
wave with a series of samples,
which is NOT a local average.
• This can be most easily shown
by Fourier transforming a 2D
grid of ones.
• The result is a single non-zero
mesh point at the center of a
mesh, but adding a guardband shows that the single
non-zero mesh point is really
an under-sampled sine
function
1
Input
FFT
FFT w GB
0.8
0.6
0.4
0.2
0
55
60
65
Matlab Script
x=1:1:128; x3=1:0.5:128.5;
E = ones(1,128);
E2 = fftshift(fft(fftshift(E)));
E3 = fftshift(fft(fftshift(addGuardBand(E',1))));
figure; plot(x,E./max(E),'-o');
hold on;
plot(x3,abs(E3)./max(abs(E3)),'g*-');
plot(x,E2./max(E2),'rx-');
zoomOut(0.1);
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70
75
One-Step Fourier Propagator
• Steps:
U '  P  Q2  F U  Q1 
– Multiply by QPF
– Fourier Transform
– Multiply by QPF &
Phase Factor
• Comments
z
2 
1
– Least computationally
expensive
– Offers no control over
the resulting mesh
spacing
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Convolution Propagator
• Steps:
– Fourier transform
– multiplication by
the Fourier
transformed kernel
– an inverse Fourier
transform
• Advantage:
U 2  P   U1  h  dx1  dy1



F (h)  H  exp  jz f x  f y
U 2  P  F 1 F h   F U1 
 P  F 1 H  F U1 
– Maintains the
mesh spacing
 2  1
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
 k
x2  x1 2   y2  y1 2 
h  exp  j
 2z

2
2

Two-Step Forward Propagator
z  z1  z 2
• Steps:
– Fourier Propagate
– Fourier Propagate
z1
z 2
i 
, 2 
1
i
• Comments:
– Easy to scale the mesh
by picking intermediate
plane location.
– Works well in long
propagation cases
because the quadratic
phase factor is applied
before the Fourier
transform
z2
 2  1
z1
δ1
δi
z1
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δ2
z2
Scaling with the Convolution Propagator
• One drawback of the convolution propagator is its
apparent inability to scale the mesh spacing in a
propagation.
• Scaling the mesh is important when propagating
with significant wavefront curvature.
• The convolution propagator can be modified to
model propagation relative to a spherical
reference wavefront curvature such that the mesh
spacing can follow the curvature of the wavefront.
• We call this Spherical-Reference Wave
Propagation or SWP.
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Spherical Reference Wave Propagation 1/3
•
•
•
•
Consider convolution
propagation with a built in
wavefront curvature relative to
the lens law.
Based on imaging, light
propagated through an ideal lens
to a distance d1 will be the same
shape as if it were propagated to
a distance d2 where f-1=d1-1+d2-1
except for a magnification term.
For example: a beam
propagating 48 cm toward a
focus through a 50cm lens.
Based on the lens law, this is
equivalent to propagating a 1200
cm without going through the
focus.
It is significantly easier to
propagate 1200 cm than it is to
propagate 48cm.
object
f
d2
d1
1 1 1
 
d 2 f d1
1
1 
 1
d2  

  1200cm
 50cm 48cm 
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image
Spherical Reference Wave Propagation 2/3
• The real advantage of
SWP is the ability to scale
the mesh to track the
evolution of the beam size.
• The spherical reference
wave radius of curvature
(Rref) can be determined
based on the desired
magnification (M).
δ1
z
Rref
1
Rref
Rref
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δ2
2
2

,M 
Rref  z
1
1
z

z
1   2
1 M
Spherical Reference Wave Propagation 3/3
• Procedure:
– Specify desired
magnification (M) and
propagation distance (z)
– Calculate effective reference
curvature (Rref)
– Determine new effective
propagation distance (zeff)
– Propagate the effective
distance and then change
the sample spacing by the
magnification factor
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2
M
1
Rref
z

1 M

1
zeff  Rref  z

1 1
Mathematical Comparison of
Convolution & Double Propagation
Convolution Propagator with SWP
Double Propagator
U 2  P  F 1 F h   F U1 
 F 1 H  F U1 
H  Qh
U 2  P  F 1 Qh  F U1 
U 2  P  Q2  F Qi  F U  Q1  Qi 
If we are propagatin g relative to a
We can combine the Qi terms to
spherical reference, Q r , this becomes :
become :
U 2  P  Q2  F Q2i  F U  Q1 
U 2  P  Qr 2  F 1 Qh  F U1  Qr1 
With the SWP addition, the Double Propagator
and the Convolution Propagator are equivalent.
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Conclusions
• We use the convolution propagator in
WaveTrain with spherical-reference wave
propagation.
• This allows us to solve any general waveoptics problem.
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Computer Modeling of Fourier
Propagation
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Computer Fourier Propagation Modeling
3
2
Phase (radians)
• In most situations,
the most rapidly
varying part of the
field is the QPF.
• In a complex field,
the phase is reset
every wavelength or
2π radians.
• To achieve proper
sampling, sampling
theory dictates that
we need two
samples per wave.
1
0
-1
-2
-3
10
20
30
40
50
60
70
Samples
Inadequate
Sampling
Adequate Sampling
k = 1; R = 16; x = -63:1:64; p = exp(j*k* (x.^2) ./ (2*R));
plot(angle(p),'b*-')
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Methods of Determining Sampling
• Diffraction Limit
• Angular Bandwidth
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Mesh Sampling:
Whittaker-Shannon Theory
• We need 2 samples
for each wave of
amplitude.
• For a parabolic
phase surface, this is
most limiting at the
edge of the phase
surface.
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λ
2δ
Mesh Sampling: Diffraction
• Diffraction is the
fundamental limit of our
ability to model
propagation.
• If we assume we need 2
samples per diffraction
spot radius on BOTH
sides of a propagation,
we get a maximum value
for the mesh spacing.
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z
D1
rspot
D2
f

 2
Dbeam
z
z
2 
and 1 
2 D1
2 D2
Mesh Sampling: Angular Bandwidth
z
 max 2
D1

z
 max 1 
D1
D1


z 2 2
D2


z
21
z
2 
2 D1
z
1 
2 D2
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D2
z
D2
Virtual Adjacent Apertures
• Now that we know the mesh
sampling intervals (δ1 and δ2),
we need to know how big a
mesh we need to use to
accurately model the
diffraction.
• The Fourier transform
assumes a repeating function
at the input.
– This means that there are
effective virtual apertures on all
sides of the input aperture.
• We need a mesh large
enough that these virtual
adjacent apertures do not
illuminate our area of interest.
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Mesh Size: Equal Sized Apertures
z
• To avoid adjacent
apertures (Dup and Ddown)
Dup
from interfering with the
output area of interest,
the modeled region
should be D+θz in
diameter.
D
D D+θz
• This means that the
number of mesh points
should be this diameter
divided by the mesh
Ddown
spacing (δ).
• For equal apertures, this
  
D

means a factor of two
 z
2
D


z
D
D
2



guard-band.
N

2 4



assuming    max  z 2 D
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z
Unequal Apertures
• For unequal apertures (such
as spherical-reference wave
propagation), the same
geometric argument can be
made.
• The resulting form can be
thought of as the average of
the number of mesh points
required to cover each of the
input apertures plus a
diffractive term.
• If we set the two mesh
spacings to their maximum
value, the number of mesh
points reduces to a familiar
form:
General form :
D1 D2
z
N


21 2 2 21 2
Minimizing N ( ' s at maximum) :
D1 D2
N 4
z
– the Fresnel number.
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Fresnel Number

z
2
• The Fresnel number
is
2a
– the number of half
waves of phase of a
parabolic wavefront
over the aperture.
– half the number of
 (a) a 2
r2
Nf 

 (r ) 
diffraction limited
 2 z
2z
spots diameters over
the aperture.
Daperture
– roughly the number of
a2
a
2a
Nf 



diffraction ripples
z z a 4 z 2 Dspot
across an aperture.
D
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Mesh Size and Fresnel Number
• For equal sized
apertures, the
number of mesh
points equals 16
times the Fresnel
number.
• For unequal
apertures, the
same is true if we
define an effective
Fresnel number as
r2r1/λz.
Equal Sized Apertures
Minimizing N ( ' s at maximum) :
D2
r2
N 4
 16  16 N f
z
z
where N f is the Fresnel number
Unequal Sized Apertures
Minimizing N ( ' s at maximum) :
D1 D2
r1r2
N 4
 16
 16 N f _ eff
z
z
where N f _ eff is the effective Fresnel number
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Summary of Mesh Determination
Mesh Size
Mesh Sample Spacing
z
z
2 
and 1 
2 D1
2 D2
r1r2
N  16 N f _ eff  16
z
for maximum 1 and  2
• Derived by either:
• Derived based on
eliminating overlap
between adjacent
virtual apertures and
the region of interest.
– Diffraction limit OR
– Required angular
bandwidth
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Conclusions
• For a simple system of two limiting
apertures, we have determined a set of
inequalities that govern the choice of the
mesh.
• Next we will look at
– how phase aberrations impact the mesh
choice, and
– how this can be extended to a system of
multiple apertures.
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Impact of Phase Aberrations on
Mesh Determination
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Sinusoidal Phase Grating
sin      N

m
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

Turbulence-Induced Aperture Blurring
• Turbulence acts to
increase the size of the
point spread function
• This effectively blurs the
apertures at each end.
• The blurred apertures can
be thought of as being
larger if we want to
capture most of the
energy.
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Gaussian
PSF
Blurred
Aperture
Blur Effect on Aperture Size
• Turbulence can be thought Blurred Edge of a Hard Aperture
of as diffracting light as if it 1
were sent through a grating
with a period equal to r0,
which is Fried’s coherence
0.5
length.
– r0 can be though of as the
characteristic turbule size.
• A scaling parameter, cturb, is
used to control the amount
of energy captured in the
calculation.
– cturb ≈ 4 is for 99% of the
energy.
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0
180
190
200
210
Deffective  Dinitial  cturb

r0
Procedure to Determine the Mesh
while Considering Turbulence
z
D1’ D1
D2
1. Simplify the turbulence distribution along the path into effective r0
values for each effective aperture.
2. Modify the aperture size using the effective r0.
3. Use the new effective aperture sizes to determine the mesh.
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D2’
Determining Fourier Propagation
Mesh Parameters for Complex
Optical Systems of Simple Optics
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Introduction
• The determination of mesh parameters for
wave-optics modeling can be uniquely
determined by a pair of limiting apertures
separated by a finite distance.
• An optical system comprised of a set of
ideal optics can be analyzed to determine
the two limiting apertures that most restrict
rays propagating through the system using
field and aperture stop techniques.
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Definitions of Field & Aperture Stop
• Aperture Stop = the aperture in a system
that limits the cone of energy from a point
on the optical axis.
• Field Stop = the aperture that limits the
angular extent of the light going through
the system.
• NOTE: All this analysis takes place in rayoptics space.
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Example System
Optical System
D=15
D=1
f=100 D=5
f=100
A1
L1
150
L2
200
Input
Input Space
D=15
Plane 2
50
Plane 3
15
15
5
1
-50
0
A1
150
A2
L1
L2
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A2
Plane 4
Procedure for Finding Stops 1/3
1.
Find the location and size of
each aperture in input space.
1.
Find the ABCD matrix from
the input of the system to
each optic in the system.
Solve for the distance (zimage)
required to drive the B term
to zero by inverting the
input-space to aperture ray
matrix.
2.
•
3.
This matrix is the mapping
from the aperture back to
input space.
The A term is the
magnification (Mimage)of the
image of that aperture.
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Mi  M
1
input space _ to _ aperture
 Ai

Ci
Bi 
Di 
0 
1 zimage    Ai Bi    M


0
   C D     C 1 / M 
1

  i
i


zimage   B D
M image  C  zimage  A
Procedure for Finding Stops 2/3
2. Find the angle formed by the edges of each of
the apertures and a point in the middle of the
object/input plane. The aperture which creates
the smallest angle is the image of the aperture
stop or the entrance pupil.
15
15
5
1
-50
0
A1
150
A2
L2
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51
L1
Procedure for Finding Stops 3/3
2. Find the aperture which most limits the
angle from a point in the center of the
image of the aperture stop in input
space. This aperture is the field stop.
15
15
5
1
-50
0
A1
150
A2
L2
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52
L1
Example: Fourier Propagation
Input Space
15
15
5
1
-50
0
A1
150
A2
L1
L2
D1 = 1 mm, D2 = 15 mm, λ = 1 μm, z = 0.15 m
Minimal Mesh = 400 x 9.375 μm = 3.75 mm
53
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Example System
Optical System
D=15
D=1
f=100 D=5
f=100
A1
L1
150
Input
L2
200
Plane 2
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D=15
A2
50
Plane 3
Plane 4
N=1024, δ=6.6 μm
Over-Sampled
Input
Plane 2 Plane 3 Plane 4
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N=512, δ=9.4 μm
Minimal
Sampling
Input
Plane 2 Plane 3 Plane 4
56
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N=256, δ=13.3 μm
Under
Sampled
Input
Plane 2 Plane 3 Plane 4
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Conclusions of Complex System Mesh
Parameter Determination
• We have devised a procedure to reduce a
complex system comprised of simple
optics into a pair of the most restricting
apertures.
• It would be nice to have a way of
simplifying a complex system of simple
optics so that modeling it is
computationally easier…
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ABCD Ray Matrix Wave-Optics
Propagator
59
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Implementation Options
U 2 x2 , y2  
exp( jkL)

j B
• Siegman combined
the ABCD terms
  Ax  y  

 jk 
directly in the
  U x , y exp  2B  2x x  y y    dx dy
 

Huygens integral.


D
x

y

 
• He then also
introduced a way of  A B   1 0 M 2 0 
C D    1 / f 1  0 1 / M  
decomposing any

 
2
2

ABCD propagation 1 L M 1 0   1 0
0 1   0 1 / M    1 / f 1 
into 5 individual


1
1

steps.
2

1
1
1
1
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1 2
2
2
60
1
1 2

2
2
2
1
1
Polishing the Siegman Decomposition
Algorithm
• We found that one of the
magnification terms was
unnecessary (M1=1.0).
• Siegman’s algorithm did not
address two important
situations: image planes
and focal planes.
• We worked a bit more on
how to pick an appropriate
magnification when
considering diffraction.
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0
A B  1
C D    1 / f 1 

 
2

0
0  1 L   1
M
 0 1 / M  0 1    1 / f 1 



1

 M
 1 / Mf

0 
1 / M 
L
D2  AD1  2
D1
Siegman Decomposition Algorithm
• Choose
magnifications M1 &
M2 (M=M1*M2)
• Calculate the
effective
propagation length
and the focal
lengths.
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A B
C D  


L
B
Leq 

2
M
M1
B
f1 
MA
B
f2 
1/ M  D
Eliminating a Magnification Term
• We determined that  A B    1 0 M 2
C D   1 / f 1  0

 
2

one of the two
magnification terms 1 L M 1 0   1
0 1   0 1 / M 1   1 / f1
that Siegman put
into his
decomposition was  A B   1
0
C D    1 / f 1 
unnecessary.

 
2

– There were five
unknowns and four
inputs.
0
1
0
0  1 L   1
M
 0 1 / M  0 1    1 / f 1 



1

63
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0 


1/ M 2 
Image Plane: B=0
• This case is an
 1
image plane.
 1 / f
2

• There is no
propagation
involved here, but
there is
– curvature and
– magnification.
Siegman
B
Leq 
0
M
B
f1 
0
MA
B
f2 
0
1/ M  D
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0 M
0 
0   M




1  0 1 / M   M / f 2 1 / M 
Our Algorithm
Leq  0
C  -1/Mf 2
1
f2 
MC
Focal Plane Case
• We were trying to
automate the
selection of the
magnification by
setting it equal to the
A term of the ABCD
matrix.
1
0

0  0


1  1 / f
Siegman, M=A
M  A0
– This minimizes the
mesh requriements
f

M
f
f
f1 

M A 0
f
f2 
0
 1
Leq 
• In doing so, we found
that the
decomposition
algorithm was
problematic at a focal
plane.
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f  1
1   1 / f
f
1 
Propagation to a Focus: A=0
1
0

f  1
1   1 / f
• For a collimated beam
going to a focus, this ray
envelope diameter is
zero.
• To handle this case, we
augmented the
magnification
determination with
diffraction.
0  0


1  1 / f
Siegman, M=A
Siegman, M=1
M  A0
M 1
f

M
f
f
f1 

M A 0
f
f2 
0
 1
B
f
M
B
f1 
f
M A
B
f2 
0
1/ M  D
Leq 
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f
1 
Leq 
Choosing Magnification while
Considering Diffraction
• We propose here to add a
diffraction term to the
magnification to avoid the
case of M=0.
• We added a tuning
parameter, η, which is the
number of effective
diffraction limited
diameters.
• We leave the selection of
magnification to the user.
L
D2  AD1  2
D1
D2
L
M
 A  2 2
D1
D1
 A
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 1
2 Nf
Common Diffraction Patterns
Airy
Normalized Intensity
Sinc
Gaussian
Normalized Radius
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Integrated Energy
Integrated Energy
Threshold = 10-10
Airy
Sinc
Gaussian
η
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We concluded that
η=5 is sufficient to
capture more than
99% of the 1D
integrated energy.
Modified Decomposition Algorithm
LM

 M f
1

 LM  1  M
 f1 f 2 Mf1 f 2
for M1  1.0
• If at an image plane
(B=0)
M=A (possible need for
interpolation)
Apply focus
• Else
Leq 
Specify M, considering
diffraction if necessary
Calculate and apply
the effective
propagation length and
the focal lengths.
L
B

2
M
M1
B
f1 
MA
B
f2 
1/ M  D
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
LM 

1 LM 
M f 2 
Implementing Negative Magnification
• After going through a focus, the
magnification is negated.
• We implement negative magnification by
inverting the field in one or both axes.
– We consider the dual axis ray matrix
propagation using the 5x5 ray matrix
formalism.
71
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Dual Axis Implementation
• In our
implementation, we
handle the case of
cylindrical telescopes
along the axes by
dividing the
convolution kernel
into separate parts for
the two axes.
U 2  P  F 1 H  F U1 
H  exp  jz
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fx  zy f y
2
x
2

Example: ABCD Propagator
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Example System
f/2
D
f
2f
2f
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Magnitude
z=2f before lens
z=2f after lens
z=2.5f
z=3f
z=3.5f
z=4f
75
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Unwrapped Phase
z=2f before lens
z=2f after lens
z=2.5f
z=3f
z=3.5f
z=4f
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ABCD Ray Matrix Fourier Propagation
Conclusions
• We have modified Siegman’s ABCD
decomposition algorithm to include several
special cases, including:
– Image planes
– Propagation to a focus
• This enables complex systems comprised
of simple optical elements to be modeled
in 4 steps.
77
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Questions?
[email protected]
(505) 245-9970 x122
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